You can calculate how much time slows aboard
a moving ship using your imagination and a little high school geometry
and algebra. You’ll end up with the same equation that Einstein
did back in 1905!
Start with Einstein’s two basic assumptions:
The speed of light is the same for all observers.
The laws of physics
are the same as long as you’re
traveling at a constant speed.
We’ll show you one way to derive the formula that lets you
calculate how the time measured by the clock aboard your speeding
spaceship differs from the time measured by an Earthbound clock.
First, use your imagination.
Imagine you have a spaceship like the one you just flew to Epsilon
Eridani 3. On this ship, you have a very simple clock. It’s
made of two mirrors, set one above the other, exactly parallel
to each other. Between these mirrors a pulse of light bounces up
and down, making the clock tick every time it hits a mirror. Doesn’t
get much simpler than that!
When the spaceship is on Earth, waiting to take off, both a person
on the ship and an observer outside the ship will agree that it
takes a certain time (let’s call it
t
_{
0
}
) for the
clock to tick. No surprise there.
The situation changes, though, when the ship flies past an Earthbound
observer. From the point of view of the person on the ship, nothing
is different: the pulse of light bounces up and down, just as it
did before, with the time
t
_{
0
}
between ticks. But here’s
what the Earthbased observer sees:
From the Earthbased observer’s point of view, the light
pulse travels on a diagonal path between the mirrors as the ship
moves forward. This diagonal path is longer than the straight upanddown
path observed by the person on the ship.
Remember that we’re assuming the speed of light is the same
for all observers. Both people observe light traveling at about
300 000 kilometers per second. But the Earthbound person observes
the light traveling a longer distance for each tick of the clock.
Since the speed of light is always the same, it must take a longer
time to travel this greater distance. Therefore the moving clock
ticks slower for an Earthbound observer than it does for an observer
on the ship. Physicists call this effect
time dilation.
Now do the math.
Here’s a diagram of the
ship moving forward.
We’ve drawn a right triangle made up of three lines:
• Line a, which is the distance the pulse of light
moves on its vertical path.
• Line b, which is the distance the ship travels in
one clock tick.
• Line d, which is the distance the pulse of light
moves on its diagonal path.
If you know the velocity of the ship and the speed of light, you
can figure out the length of each line in the triangle. All you
need to know is how to convert time and speed to distance. You
probably know that if you travel 60 kilometers per hour for 2 hours,
you’ve traveled 120 kilometers. You calculate the distance
by multiplying velocity by time: 60 kilometers per hour times 2
hours equals 120 kilometers.
Now that you know how to calculate distance, you can figure out
the distances in the triangle. Line a, the distance the light travels
vertically, is the velocity (the speed of light, or
c
)
multiplied by the tick time
t
_{
0
}
that’s observed by
a person on the ship:
a = distance the light travels
vertically =
ct
_{
0
}
.
Line b, the distance the ship travels in one tick of the clock
(
t,
as measured by the Earthbound observer), is velocity
v
multiplied
by the time the clock took to tick
t:
b = distance the ship travels
=
vt.
And line d, the distance the pulse of light moves on its diagonal
path, is the velocity (again, the speed of light, or
c
)
multiplied by the time it took the clock to tick as measured by
the Earthbound observer. That’s
d = distance the light travels
diagonally =
ct.
Now we want to find a relationship between
t
and
t
_{
0
}
.
The Pythagorean theorem, which you learned in geometry, shows the
relationship among the sides of a right triangle.
According to the Pythagorean theorem,
d
^{
2
}
= a
^{
2
}
+
b
^{
2
}
Now you can substitute in the distances you calculated above. You get
(
ct
)
^{
2
}
= (
ct
_{
0
}
)
^{
2
}
+ (
vt
)
^{
2
}
With a fair amount of
algebraic
maneuvering
, you can change this equation into
Finally, we have an equation that tells us how long the clock
will take to tick
t
given the timeaboardship tick
t
_{
0
}
and the speed of the ship
v.
The speed of light
c
is
a constant.
What happens as the velocity of the ship
v
approaches
the speed of light
c
? The quantity
v
^{
2
}
gets closer
to the quantity
c
^{
2
}
. Therefore
v
^{
2
}
/
c
^{
2
}
approaches 1 and (1 –
v
^{
2
}
/
c
^{
2
}
) approaches
0. So does its square root. Anything divided by a number that approaches
zero gets very large and approaches infinity. And in our equation,
the shipboardtime (
t
_{
0
}
) is being divided by a number that’s
approaching zero. So, as viewed from Earth, the faster the ship
moves, the longer the clock takes to tick (
t
).
Another way to write this equation is
You can separate the right side of this equation into two parts—time
and a quantity that physicists call
gamma
(
):
Then you can write the time dilation equation as
t
=
t
_{
0
}
.
You’ll find gamma in other calculations related to special
relativity. You’ll see it in the equations that show how
mass increases and length contracts with increasing speed. Once
you know the value of gamma, the math for time dilation and other
aspects of special relativity becomes very simple.
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Thinking Like Einstein
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